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Validating smartphone sensor-based tests to study gait and balance against reference measurement systems in a laboratory setting poses several technical challenges related to data quality and data processing. One challenge is to guarantee the correct annotation of the data, which is required to ensure that only data collected during the same test execution are compared across measurement systems in subsequent analyses. A second challenge is to accurately synchronize the data across the different systems. Here, we propose innovative solutions for both challenges and illustrate their use in the example of comparing smartphone sensor data collected with the Floodlight technology with data collected with a motion capture system. These solutions form important tools for guaranteeing the data quality and data integrity required for the validation of gait and balance characteristics measured by digital health technology tools such as the Floodlight technology., Comment: 16 pages, 4 figures, 1 table

In this paper, we present a novel approach for learning bimanual manipulation actions from human demonstration by extracting spatial constraints between affordance regions, termed affordance constraints, of the objects involved. Affordance regions are defined as object parts that provide interaction possibilities to an agent. For example, the bottom of a bottle affords the object to be placed on a surface, while its spout affords the contained liquid to be poured. We propose a novel approach to learn changes of affordance constraints in human demonstration to construct spatial bimanual action models representing object interactions. To exploit the information encoded in these spatial bimanual action models, we formulate an optimization problem to determine optimal object configurations across multiple execution keypoints while taking into account the initial scene, the learned affordance constraints, and the robot's kinematics. We evaluate the approach in simulation with two example tasks (pouring drinks and rolling dough) and compare three different definitions of affordance constraints: (i) component-wise distances between affordance regions in Cartesian space, (ii) component-wise distances between affordance regions in cylindrical space, and (iii) degrees of satisfaction of manually defined symbolic spatial affordance constraints., Comment: 8 pages, accepted for publication at Humanoids 2024 - Copyright IEEE

We propose a new fast algorithm for simultaneous recovery of the coil sensitivities and of the magnetization image from incomplete Fourier measurements in parallel MRI. Our approach is based on a parameter model for the coil sensitivities using bivariate trigonometric polynomials of small degree. The derived MOCCA algorithm has low computational complexity of $O(N_c N^2 \log N)$ for $N \times N$ images and $N_c$ coils and achieves very good performance for incomplete MRI data. We present a complete mathematical analysis of the proposed reconstruction method. Further, we show that MOCCA achieves similarly good reconstruction results as ESPIRiT with a considerably smaller numerical effort which is due to the employed parameter model. Our numerical examples show that MOCCA can outperform several other reconstruction methods., Comment: 32 pages

In this paper we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We extend the differential approximation method proposed in [4,39] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$ in the weighted space $L_2({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})$ where $\sigma, \, \rho >0$. We prove that the optimal frequency parameters $\lambda_1, \ldots , \lambda_{N}$ for this method in the approximation problem $ \min\limits_{\lambda_{1},\ldots, \lambda_{N}, \gamma_{1} \ldots \gamma_{N}}\|\mathrm{e}^{-\cdot^{2}/2\sigma} - \sum\limits_{j=1}^{N} \gamma_{j} \, {\mathrm e}^{\lambda_{j} \cdot}\|_{L_{2}({\mathbb R}, \mathrm{e}^{-t^{2}/2\rho})}$, are zeros of a scaled Hermite polynomial. This observation leads us to a numerically stable approximation method with low computational cost of $\mathit{O}(N^{3})$ operations. Furthermore, we derive a direct algorithm to solve this approximation problem based on a matrix pencil method for a special structured matrix. The entries of this matrix are determined by hypergeometric functions. For the weighted $L_{2}$-norm, we prove that the approximation error decays exponentially with respect to the length $N$ of the sum. An exponentially decaying error in the (unweighted) $L^{2}$-norm is achieved using a truncated cosine sum., Comment: 30 pages, 12 figures

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to the model of continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give a recursive algorithm to transfer the parameter set determining the ReLU DNN into the parameter set of a CPL spline function. Using this representation, we show that after removing the well-known parameter redundancies of the ReLU DNN, which are caused by the positive scaling property, all remaining parameters are independent. Moreover, we show that the ReLU DNN with one, two or three hidden layers can represent CPL spline functions with $K$ arbitrarily prescribed knots (breakpoints), where $K$ is the number of real parameters determining the normalized ReLU DNN (up to the output layer parameters). Our findings are useful to fix a priori conditions on the ReLU DNN to achieve an output with prescribed breakpoints and function values., Comment: 31 pages, 5 figurs

In this paper we introduce two new algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz+Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case., Comment: 29 pages, 6 figures

We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the AAA algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices which are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the matrix pencil method is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the matrix pencil method but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the matrix pencil method for noisy data and for signal approximation by short exponential sums., Comment: 36 pages, 18 figures

In this paper we derive a new recovery procedure for the reconstruction of extended exponential sums of the form $y(t) = \sum_{j=1}^{M} \left( \sum_{m=0}^{n_j} \, \gamma_{j,m} \, t^{m} \right) {\mathrm e}^{2\pi \lambda_j t}$, where the frequency parameters $\lambda_{j} \in {\mathbb C}$ are pairwise distinct. For the reconstruction we employ a finite set of classical Fourier coefficients of $y$ with regard to a finite interval $[0,P] \subset {\mathbb R}$ with $P>0$. Our method requires at most $2N+2$ Fourier coefficients $c_{k}(y)$ to recover all parameters of $y$, where $N:=\sum_{j=1}^{M} (1+n_{j})$ denotes the order of $y$. The recovery is based on the observation that for $\lambda_{j} \not\in \frac{{\mathrm i}}{P} {\mathbb Z}$ the terms of $y$ possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [12]. If a sufficiently large set of $L$ Fourier coefficients of $y$ is available (i.e., $L > 2N+2$), then our recovery method automatically detects the number $M$ of terms of $y$, the multiplicities $n_{j}$ for $j=1, \ldots , M$, as well as all parameters $\lambda_{j}$, $j=1, \ldots , M$ and $ \gamma_{j,m}$ $j=1, \ldots , M$, $m=0, \ldots , n_{j}$, determining $y$. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method., Comment: 29 pages, 7 figures

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form $f(t) = \sum_{j=1}^{K} \gamma_{j} \, \cos(2\pi a_{j} t + b_{j})$, where the frequency parameters $a_{j} \in {\mathbb R}$ (or $a_{j} \in {\mathrm i} {\mathbb R}$) are pairwise different. Our method is based on the recently proposed stable iterative rational approximation algorithm in \cite{NST18}. For signal reconstruction we use a set of classical Fourier coefficients of $f$ with regard to a fixed interval $(0, P)$ with $P>0$. Even though all terms of $f$ may be non-$P$-periodic, our reconstruction method requires at most $2K+2$ Fourier coefficients $c_{n}(f)$ to recover all parameters of $f$. We show that in the case of exact data, the proposed iterative algorithm terminates after at most $K+1$ steps. The algorithm can also detect the number $K$ of terms of $f$, if $K$ is a priori unknown and $L>2K+2$ Fourier coefficients are available. Therefore our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method. Keywords: sparse exponential sums, non-harmonic Fourier sums, reconstruction of sparse non-periodic signals, rational approximation, AAA algorithm, barycentric representation, Fourier coefficients, Comment: 28 pages, 2 figures

Internet users and service providers do not often know when traffic is being watched but desire a way to determine when, where, and by whom. We present NOISE, the Nonce Observatory for Inverse Surveillance of Eavesdroppers, a method and system that detects monitoring by disseminating nonces - unique, pseudorandom values - in traffic and seeing if they are acted upon unexpectedly, indicating that the nonce-laden traffic is being monitored. Specifically, we embed 64-bit nonces innocuously into IPv6 addresses and disseminate these nonces Internet-wide using a modified traceroute-like tool that makes each outbound probe's source address unique. We continually monitor for subsequent nonce propagation, i.e., activity or interest involving these nonces, e.g., via packet capture on our system's infrastructure. Across three experiments and four months, NOISE detects monitoring more than 200k times, ostensibly in 268 networks, for probes destined for 437 networks. Our results reveal: (a) data collection for security incident handling, (b) traffic information being shared with third parties, and (c) eavesdropping in or near a large commercial peering exchange.

In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (2018) for fast reconstruction of $M$-sparse vectors ${\mathbf x}$ of length $N= 2^J$, where we assume that all components of the discrete Fourier transform $\hat{\mathbf x}= {\mathbf F}_{N} {\mathbf x}$ are available. The sparsity of ${\mathbf x}$ needs not to be known a priori, but is determined by the algorithm. If the sparsity $M$ is larger than $2^{J/2}$, then the algorithm turns into a usual FFT algorithm with runtime ${\mathcal O}(N \log N)$. For $M^{2} < N$, the runtime of the algorithm is ${\mathcal O}(M^2 \, \log N)$. The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $M>20$ because of numerical instabilities, the modified algorithm is still numerically stable for $M=200$., Comment: 15 pages, 3 figures

We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm., Comment: 34 pages

We present the result of an experiment to measure the electric dipole moment (EDM) of the neutron at the Paul Scherrer Institute using Ramsey's method of separated oscillating magnetic fields with ultracold neutrons (UCN). Our measurement stands in the long history of EDM experiments probing physics violating time reversal invariance. The salient features of this experiment were the use of a Hg-199 co-magnetometer and an array of optically pumped cesium vapor magnetometers to cancel and correct for magnetic field changes. The statistical analysis was performed on blinded datasets by two separate groups while the estimation of systematic effects profited from an unprecedented knowledge of the magnetic field. The measured value of the neutron EDM is $d_{\rm n} = (0.0\pm1.1_{\rm stat}\pm0.2_{\rm sys})\times10^{-26}e\,{\rm cm}$., Comment: 5 pages, 4 figures, submitted to PRL on 18.12.2019

In this survey we describe some modifications of Prony's method. In particular, we consider the recovery of general expansions into eigenfunctions of linear differential operators of first order and show, how these expansions can be recovered from function samples using generalized shift operators. We derive an ESPRIT-like algorithm for the generalized recovery method and show, that this approach can be directly used to reconstruct classical exponential sums from non-equispaced data. Furthermore, we derive a modification of Prony's method for sparse approximation with exponential sums which leads to a non-linear least-squares problem., Comment: 27 pages, 2 figures